Abstract - Brown University · This is why the schemes are called “upwind-biased”. Also notice...

Stability analysis of the inverse Lax-Wendroff boundary treatment for high

order upwind-biased finite difference schemes

Tingting Li1, Chi-Wang Shu2 and Mengping Zhang3

Abstract

In this paper, we consider linear stability issues for one-dimensional hyperbolic con-

servation laws using a class of conservative high order upwind-biased finite difference

schemes, which is a prototype for the weighted essentially non-oscillatory (WENO)

schemes, for initial-boundary value problems (IBVP). The inflow boundary is treated

by the so-called inverse Lax-Wendroff (ILW) or simplified inverse Lax-Wendroff (SILW)

procedure, and the outflow boundary is treated by the classical high order extrapola-

tion. A third order total variation diminishing (TVD) Runge-Kutta time discretization

is used in the fully discrete case. Both GKS (Gustafsson, Kreiss and Sundstrom) and

eigenvalue analysis are performed for both semi-discrete and fully discrete schemes. The

two different analysis techniques yield consistent results. Numerical tests are performed

to demonstrate the stability results predicted by the analysis.

Key Words: high order upwind-biased schemes; inverse Lax-Wendroff procedure; sim-

plified inverse Lax-Wendroff procedure; extrapolation; stability; GKS theory; eigenvalue

analysis.

1School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui

230026, P.R. China. E-mail: [email protected] of Applied Mathematics, Brown University, Providence, RI 02912, USA. E-mail:

[email protected]. Research supported by AFOSR grant F49550-12-1-0399 and NSF grant DMS-

1418750.3School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui

230026, P.R. China. E-mail: [email protected]. Research supported by NSFC grant 11471305.

1

1 Introduction

When a high order finite difference scheme with wide stencil is used to solve hyper-

bolic conservation laws, the inner schemes cannot be used near the boundary. Special

treatments near the boundaries are needed in order to maintain accuracy and stabil-

ity. There exist two difficulties when imposing numerical boundary conditions. Firstly,

the points used in these schemes which lie outside the computational domain, namely

the “ghost points”, should be evaluated properly. Secondly, the grid points may not

coincide with the physical boundary exactly. For hyperbolic conservation laws, classi-

cal Lagrangian extrapolation to evaluate ghost point values near the outflow boundary

usually leads to stable approximations. However, it is a challenge to obtain stable and

accurate numerical boundary conditions near the inflow boundary. This is especially

the case when the physical boundary does not coincide with but is very close to the

first grid point, which is referred to as the “cut-cell” problem in the literature, see e.g.

[2]. The inverse Lax-Wendroff (ILW) procedure, first introduced in [16], can overcome

this difficulty. The simplified ILW (SILW) procedure, which is an extension of the ILW

procedure and can save in algorithm complexity and computational cost, is introduced

in [18]. For earlier related work, see [5, 6, 9, 8, 10].

Stability of the numerical schemes for initial boundary value problems (IBVP) on

finite domain can be established by the normal mode analysis, which is based on the

Laplace transform. General stability analysis based on this technique is the famous

Gustafsson, Kreiss and Sundstrom (GKS) theory [7]. In [7], stability of fully discrete

finite difference schemes is analyzed. In [15], stability analysis is performed for the semi-

discrete cases. For the ILW and SILW procedures when high order central compact

spatial operations are used, such GKS analysis has been performed in [19]. However,

the GKS analysis may lead to high algebraic complexity in the case of very high or-

der accuracy. An alternative technique, by visualizing the eigenvalue spectrum of the

discretization operators, has also been used in [19] to analyze stability. It has been

2

observed in [19] that, when both techniques are used, they produce consistent stability

conclusions. In this paper, we are interested in studying stability of semi-discrete and

fully discrete upwind-biased high order finite difference schemes, which serve as a pro-

totype for the weighted essentially non-oscillatory (WENO) schemes [12, 11, 13]. ILW

and SILW procedures will be used near the inflow boundary and classical Lagrangian

extrapolation will be used near the outflow boundary. Both the GKS analysis and the

eigenvalue analysis will be performed.

This paper is organized as follows. In Section 2, we review high order upwind-biased

finite difference methods as the inner schemes, and the third order total variation dimin-

ishing (TVD) Runge-Kutta time discretization method used in the full discretization.

The ILW procedure, the SILW procedure and classical extrapolation are introduced in

detail also in this section. Stability analysis is performed in Section 3 by using the GKS

theory and the eigenvalue spectrum method, first for the semi-discrete case and then for

the fully discrete case. Numerical tests are provided in Section 4 to demonstrate the

results of the analysis. Concluding remarks are given in Section 5.

2 Scheme formulation

In this section, we review high order upwind-biased finite difference methods as the

inner schemes, and the third order total variation diminishing (TVD) Runge-Kutta time

discretization method used in the full discretization. We also introduce the ILW pro-

cedure, the SILW procedure and classical extrapolation used in the inflow and outflow

boundary treatments.

2.1 High order upwind-biased finite difference schemes

Consider the one-dimensional scalar conservation law

ut + f(u)x = 0, x ∈ [a, b], t ≥ 0

u(a, t) = g(t), t ≥ 0

u(x, 0) = u0(x), x ∈ [a, b]

(2.1)

3

Assume that f ′(u(a, t)) > 0 and f ′(u(b, t)) > 0 for t > 0. This assumption guarantees

that the left boundary x = a is an inflow boundary where a boundary condition is needed,

and the right boundary x = b is an outflow boundary where no boundary condition can

be prescribed.

The interval (a, b) is discretized by an uniform mesh as

a + Ca ∆x = x0 < x1 < x2 < · · · < xN = b − Cb ∆x (2.2)

where Ca ∈ [0, 1) and Cb ∈ [0, 1). {xj = a + (Ca + j) ∆x, j = 0, 1, 2, · · ·N} are the grid

points. The first and the last grid points are not necessarily aligned with the boundary,

and we choose this kind of discretization on purpose.

The general semi-discrete conservative finite difference scheme approximating (2.1),

based on point values and numerical fluxes, is of the form:

duj

dt= − 1

∆x(fj+ 1

2(t) − fj− 1

2(t)) (2.3)

where the numerical flux fj+ 12

is defined as a linear combination of f(u(x, t)) in the

neighborhood of xj such that the right hand of (2.3) approximates −f(u)x at x = xj to

the desired order of accuracy.

For convenience, the semi-discrete approximation (2.3) can be written as

duj

dt= − 1

∆xDj,k,m · fj ≡ − 1

∆x

m∑

l=0

dk,l fj−k+l (2.4)

where j − k denotes the left most point of the derivative stencil having m + 1 points.

Both k and m depend on the order of the scheme.

Schemes considered in this paper are listed below.

• Third order scheme

duj

dt= − 1

∆x(1

6fj−2 − fj−1 +

1

2fj +

1

3fj+1)

• Fifth order scheme

duj

dt= − 1

∆x(− 1

30fj−3 +

1

4fj−2 − fj−1 +

1

3fj +

1

2fj+1 −

1

20fj+2)

4

• Seventh order scheme

duj

dt= − 1

∆x(

1

140fj−4 −

7

105fj−3 +

3

10fj−2 − fj−1

+1

4fj +

3

5fj+1 −

1

10fj+2 +

1

105fj+3)

• Ninth order scheme

duj

dt= − 1

∆x( − 1

630fj−5 +

1

56fj−4 −

2

21fj−3 +

1

3fj−2 − fj−1

+1

5fj +

2

3fj+1 −

1

7fj+2 +

1

42fj+3 −

1

504fj+4)

• Eleventh order scheme

duj

dt= − 1

∆x(

1

2772fj−6 −

1

210fj−5 +

5

168fj−4 −

5

42fj−3 +

5

14fj−2 − fj−1

+1

6fj +

5

7fj+1 −

5

28fj+2 +

5

126fj+3 −

1

168fj+4 +

1

2310fj+5)

• Thirteenth order scheme

duj

dt= − 1

∆x( − 1

12012fj−7 +

1

792fj−6 −

1

110fj−5 +

1

24fj−4 −

5

36fj−3 +

3

8fj−2 − fj−1

+1

7fj +

3

4fj+1 −

5

24fj+2 +

1

18fj+3 −

1

88fj+4 +

1

660fj+5 −

1

10296fj+6)

Notice that all schemes have one more point on the left than on the right for their

stencils, considering the positive wind direction. This is why the schemes are called

“upwind-biased”. Also notice that these schemes are just the standard WENO schemes

with the linear weights (when the smoothness indicators and nonlinear weights are turned

off), see [1].

2.2 Time discretization

We use a third order TVD Runge-Kutta method [14] to integrate the semi-discrete

system of ordinary differential equations (ODEs) (2.3) in time. For simplicity, the system

of the initial value problems of ODEs is written as

ut = Lu

5

From the time level tn to tn+1, the third order TVD Runge-Kutta method is given by

u(1) = un + ∆tL(un)

u(2) =3

4un +

1

4u(1) +

1

4∆tL(u(1))

un+1 =1

3un +

2

3u(2) +

2

3∆tL(u(2))

where ∆t is the time step.

Special attention must be taken when we impose time dependent boundary condi-

tions in the two interior stages of the Runge-Kutta method. With the time dependent

boundary condition g(t), the traditional match of time is

un ∼ g(tn)

u(1) ∼ g(tn + ∆t)

u(2) ∼ g(tn +∆t

2)

but this would decrease the accuracy to second order as pointed out in [3]. So we use

the following match of time which is analyzed in [3] to ensure third order accuracy in

the time discretization

un ∼ g(tn)

u(1) ∼ g(tn) + ∆tg′(tn)

u(2) ∼ g(tn) +1

2∆tg′(tn) +

1

4∆t2g′′(tn)

2.3 The inverse Lax-Wendroff (ILW) procedure

The basic idea of the ILW procedure is to use Taylor expansion at the boundary point

and then repeatedly use the PDE and its time derivatives to convert spatial derivatives

to time derivatives, in order to obtain accurate values at the relevant ghost points. The

procedure is summarized as follows.

A Taylor expansion at the boundary point a gives

u(x−p) = u(a + (Ca − p)∆x) =d−1∑

k=0

u∗(k)(∆x)k(−p + Ca)k

k!+ O(∆xd)

6

where u(x−p) is the value of the function u at the ghost point x−p. Clearly,

u−p =

d−1∑

k=0

u∗(k)(∆x)k(−p + Ca)k

k!(2.5)

is a d-th order approximation of u(x−p), if u∗(k) is an (at least) (d − k)-th order ap-

proximation of ∂ku∂xk |x=a. For the ILW procedure, the values of u∗(k) can be obtained as

follows

u∗(0) = u(a, t) = g(t)

u∗(1) =∂u

∂x|x=a = − g′(t)

f ′(g(t))

u∗(2) =∂2u

∂x2|x=a =

f ′(g(t))g′′(t) − 2f ′′(g(t))(g′(t))2

(f ′(g(t)))3

u∗(3) =∂3u

∂x3|x=a

=9f ′(g(t))f ′′(g(t))g′(t)g′′(t) + 3f ′(g(t))f ′′′(g(t))(g′(t))3

(f ′(g(t)))5

+−(f ′(g(t)))2g′′′(t) − 12(f ′′(g(t)))2(g′(t))3

(f ′(g(t)))5

· · ·

In this way, we can get u∗(k), k = 0, 1, 2 · · ·d − 1, through the given boundary data

g(t) and its time derivatives. Plugging them into (2.5), we can obtain u−p.

2.4 The simplified inverse Lax-Wendroff (SILW) procedure

The ILW procedure outlined in the previous subsection can be easily verified for sta-

bility through the GKS theory, both for the semi-discrete version [16] and for the fully

discrete version. However, for multi-dimensional nonlinear PDE systems, the derivation

of u∗(k) through the ILW procedure can be algebraically very complicated. This is es-

pecially the case for problems with moving boundaries [17]. A simplified version of the

ILW procedure, which is referred to as SILW, is introduced in [18] to save in algorithm

complexity and computational cost. This SILW procedure is analyzed for its stability in

[19] for spatially central compact discretizations.

7

The SILW procedure also uses (2.5) to obtain u−p. For k ≤ kd − 1, u∗(k) is obtained

in the same way as in the ILW procedure. For k ≥ kd, u∗(k) is obtained by the classical

Lagrangian extrapolation as u∗(k) = ∂kPd

∂xk |x=a, where Pd(x) is an interpolation polynomial

of order d using points inside the computational domain. We refer to the next subsection

for the details of the construction of Pd(x). Clearly, the choice of the threshold index

kd will be a key issue to ensure stability, which will be discussed in detail later in this

paper.

It would appear that it might benefit to use the partial information of u∗(k) for

k ≤ kd − 1, which are obtained through the ILW procedure using the given boundary

data g(t) and its derivatives, when constructing the polynomial Pd(x), so that fewer

points inside the computational domain need to be used. However, it turns out that this

would make the scheme less stable [19]. Therefore, in this paper we consider only the

pure interpolation polynomial Pd(x) without using the boundary data g(t), for computing

u∗(k) with k ≥ kd.

2.5 Extrapolation

In order to obtain uN+p for the ghost points near the outflow boundary, we use the d

points inside the computational domain, {(xN , uN), (xN−1, uN−1), · · · , (xN−d+1, uN−d+1)}

to obtain a classical interpolation polynomial Pd(x) of degree d − 1, and then take

uN+p = Pd(xN+p) to approximate u(xN+p) with is accurate of order d. That is,

uN+p =

d∑

i=1

βi uN+1−i

where {βi, i = 1, 2, · · · , d} are the coefficients defined by

βi =

d∏

t=1,t6=i

t + p − 1

t − i

The complete scheme of the numerical approximation to the one-dimensional scalar

conservation law (2.1) is as follows. One of the schemes in Section 2.1 is used as the inner

scheme, which is used everywhere inside the computational domain. The necessary ghost

8

point values are obtained as below. Ghost point values of u near the inflow boundary

are obtained by the ILW procedure or the SILW procedure introduced in Section 2.3

and Section 2.4 respectively, and ghost point values of u near the outflow boundary

are obtained by extrapolation introduced in Section 2.5. The third order TVD Runge-

Kutta method introduced in Section 2.2 is used as the time discretization. Of course,

the specific third order TVD Runge-Kutta method is used here only as an example, and

other time discretization methods can also be used.

3 Stability analysis

Stability analysis for the semi-discrete schemes and the fully discrete schemes are

discussed in this section. Both GKS and eigenvalue spectrum visualization methods

are used. Here, the analysis is performed on the one-dimensional linear IBVP (2.1)

corresponding to f(u) = cu with c > 0 and the third order TVD Runge-Kutta scheme.

3.1 The semi-discrete schemes

In this subsection we discuss semi-discrete schemes.

3.1.1 The GKS analysis

The procedure of the GKS analysis for the semi-discrete schemes is:

(1) Divide the problem into the summation of three simpler problems: one Cauchy

problem on (−∞, +∞) and two quarter-plane problems on the domains [a, +∞)

and (−∞, b], respectively.

(2) For the Cauchy problem, stability can be obtained by standard Fourier analysis.

The following analysis is performed under the assumption that the inner scheme

corresponding to the Cauchy problem is stable.

(3) For a quarter-plane problem, a necessary and sufficient condition for stability is that

there exists no eigensolution (to be explained later).

9

Take uj(t) = estφj, (2.4) becomes

∆xsφj + c(Dj,k,m · φj) = 0 (3.6)

{φj(s)}∞0 is an eigensolution if it satisfies the following constraints:

(1) It is not identically 0.

(2) It satisfies (3.6) at all points where it is applied and the numerical boundary condi-

tions.

(3) Re(s) ≥ 0.

(4) For Re(s) > 0, the corresponding solution satisfies (1) and (2) and

limj→∞

φj(s) = 0 (3.7)

(5) For Re(s) = 0, the corresponding solution should satisfy (1), (2) and (4) with respect

to s0 where

s0 = limǫ→0+

(s + ε)

s is regarded as an eigenvalue and the corresponding {φj(s)}∞0 as an eigensolution. If

an eigenvalue and the associated eigensolution exist, the scheme is unstable. Otherwise,

the scheme is stable.

Firstly, stability analysis is performed on the right-quarter plane problem:

ut + c ux = 0, x ∈ [a, +∞], t ≥ 0, c > 0

u(a, t) = g(t), t ≥ 0

u(x, 0) = u0(x), x ∈ [a, +∞]

(3.8)

For the purpose of stability analysis, g(t) can be set to zero without loss of generality.

We take first the third order inner scheme to approximate (3.8), which can be written

as

duj

dt= − c

∆x(1

6uj−2 − uj−1 +

1

2uj +

1

3uj+1), j = 2, 3, · · · (3.9)

10

Let uj = estφj , (3.9) can be transformed into

sφj = −(1

6φj−2 − φj−1 +

1

2φj +

1

3φj+1) (3.10)

Here s = s∆xc

. s can also be regarded as the eigenvalue and {φj(s)}∞0 is the corresponding

eigensolution.

The characteristic equation is obtained by taking φj = κj in (3.10):

sκ2 = −(1

6− κ +

1

2κ2 +

1

3κ3) (3.11)

Let

f(κ) = sκ2 + (1

6− κ +

1

2κ2 +

1

3κ3)

Take κ = eıξ, ξ ∈ [0, 2π]. This means |κ| = 1. From (3.11), we can get

s = −1

6e−2ıξ + e−ıξ − 1

2− 1

3eıξ

Figure 3.1 shows the locus of s for |κ| = 1 (ξ ∈ [0, 2π]).

-1.2-1.0-0.8-0.6-0.4-0.2Re@\tildeHsLD

-1.0

-0.5

0.5

1.0

Im@\tildeHsLD

Figure 3.1: Locus of s

Figure 3.1 shows that, if |κ| = 1, Re(s) ≤ 0.

Since the domain here is [a, +∞), the limit in (3.7) is j → +∞. We are only interested

in the roots of the characteristic equation which satisfy |κ| < 1.

11

If Re(s) > 0, Figure 3.1 also implies that the number of roots with |κ| < 1 of

the characteristic equation is independent of s. One can choose any s which satisfies

Re(s) > 0 to get the number of roots within |κ| < 1. Taking s = 1, the roots of (3.11)

are

κ1 = 0.303322 + 0.076861ı, κ2 = 0.303322 − 0.076861ı, κ3 = −5.10664

So, if Re(s) > 0, there are two roots satisfying |κ| < 1.

If Re(s) = 0, s = hı. One can find that (i) h ∈ [0, hmax] (here hmax is about 0.1),

the three roots of the characteristic equation satisfy |κ1| = 1, |κ2| < 1, |κ3| > 1. (ii)

h ∈ [hmax, +∞), the three roots satisfy |κ1| < 1, |κ2| < 1 and |κ3| > 1.

If |κ1| = 1, a perturbation analysis is used to decide whether the scheme is stable.

For instance, the roots of the characteristic equation with s = 0 are κ1 = 1, κ2 =

14(−5 +

√33), κ3 = 1

4(−5 −

√33). We have to determine whether κ1 = 1 is stable to

perturbation or not. To do so, we substitute s = δ and κ = 1+ǫ into the the characteristic

equation (3.11) and obtain δ = −(1/6 − (1 + ǫ) + (1 + ǫ)2/2 + (1 + ǫ)3/3)/(1 + ǫ)2.

Taylor expansion at ǫ = 0 gives δ = −ǫ + O(ǫ2). If δ > 0 which indicates Re(s) > 0,

|κ| = |1 + ǫ| < 1. κ = 1 is thus stable under perturbation.

In conclusion, for any value of s satisfying Re(s) ≥ 0, the roots of the characteristic

equation satisfy |κ1| ≤ 1, |κ2| < 1, |κ3| > 1 and κ1 = 1 is stable under perturbation.

Case 1. If κ1 and κ2 are distinct, the general expression of φj is:

φj = σ1κj1 + σ2κ

j2 (3.12)

σ1, σ2 are two constants which remain to be determined by the numerical boundary

conditions.

For the ILW boundary condition, the numerical boundary conditions are (3.10) with

j = 0, 1 with the values of the ghost points obtained through the ILW procedure. As

g(t) = 0, u−1 = 0, u−2 = 0 and φ−1 = 0, φ−2 = 0. The numerical boundary conditions

12

become:

sφ0 = −(1

2φ0 +

1

3φ1)

sφ1 = −(−φ0 +1

2φ1 +

1

3φ2)

(3.13)

Putting (3.12) into (3.13), one can get a linear system Aσ = 0 with σ = (σ1, σ2)T . The

coefficient matrix is

A =

s + 12

+ 13κ1 s + 1

2+ 1

3κ2

−1 + (s + 12)κ1 + 1

3κ2

1 −1 + (s + 12)κ2 + 1

3κ2

2

In order to get nontrivial φj , σ1 and σ2 cannot be equal to zero at the same time.

The determinant of the matrix A then must equal to 0. {s, κ1, κ2} are obtained by

{f(κ1) = 0, f(κ2) = 0, det(A) = 0, κ1 6= κ2}. The system has no solution.

Case 2. If κ1 = κ2 = κ, the general expression of φj is:

φj = σ1κj + σ2jκ

j (3.14)

The numerical boundary conditions are (3.13). Putting (3.14) into (3.13), one can get a

linear system Aσ = 0. The coefficient matrix is

A =

s + 12

+ 13κ 1

3κ

−1 + (s + 12)κ + 1

3κ2 (s + 1

2)κ + 2

3κ2

here κ is a multiple root of the equation f(κ) = 0 and it satisfies {f(κ) = 0, f′

(κ) = 0}.

Solving this system, one can get the roots:

{s = 0.890809, κ = 0.322185}, {s = −0.445405− 1.15981ı, κ = −0.161093 + 1.75438ı},

{s = −0.445405 + 1.15981ı, κ = −0.161093 − 1.75438ı}

The root satisfying Re(s) > 0 and |κ| < 1 is {s = 0.890809, κ = 0.322185}. Putting

it into Aσ = 0, we obtain det(A) = 0.830576 and {σ1 = 0, σ2 = 0}. Thus it is not an

eigensolution.

In conclusion, the scheme (3.8) with the third order inner scheme and the ILW proce-

dure boundary treatment has no eigensolution, hence it is stable. In fact, since the ghost

13

point values are completely determined by the boundary data g(t) and its derivatives

and do not depend on the numerical solution inside the computational domain, when

g(t) = 0 all the ghost point values are zero. Hence stability of the IBVP for the scheme

is the same as that for the inner scheme without boundary. That is, stability for the

ILW boundary condition is the same as that for the Cauchy problem of the inner scheme

[16].

Next, we use the SILW procedure to obtain values of ghost points and repeat the

previous analysis. In this case, the values of u−1, u−2 are functions of Ca and they

change with different choices of kd. Hence, the coefficient matrix of the linear system

for {σ1, σ2} is a function of Ca and changes with kd, too. If kd = 1, derivatives at the

boundary point x = a are:

u∗(0) = 0

u∗(1) = −(3 + 2Ca)u0 − 4(1 + Ca)u1 + (1 + 2Ca)u2

2∆x

u∗(2) =u0 − 2u1 + u2

(∆x)2

Putting this into (2.5), we obtain

φ−1 = (−C2a

2− 3

2Ca + 2)φ0 + (C2

a + 2Ca − 3)φ1 + (−C2a

2− Ca

2+ 1)φ2

φ−2 = (−C2a

2− 3

2Ca + 5)φ0 + (C2

a + 2Ca − 8)φ1 + (−C2a

2− Ca

2+ 3)φ2

(3.15)

The numerical boundary conditions are

1

12(−8 + 15Ca + 5C2

a + 12s)φ0 +1

6(12 − 10Ca − 5C2

a)φ1 +1

12(−6 + 5Ca + 5C2

a)φ2 = 0

1

12(−8 − 3Ca − C2

a)φ0 +1

12(4Ca + 2C2

a + 12s)φ1 +1

12(6 − Ca − C2

a)φ2 = 0

(3.16)

The characteristic equation is (3.11).

Case 1. If κ1 and κ2 are different. φj is in the form (3.12). Putting it into (3.16),

we can get a linear system Aσ = 0 for {σ1, σ2} with the coefficient matrix

A =

f1(κ1) f1(κ2)

f2(κ1) f2(κ2)

14

where

f1(κ) = −2

3+

5

12C2

a(κ − 1)2 + 2κ − 1

2κ2 +

5

12Ca(3 − 4κ + κ2) + s

f2(κ) = −2

3− 1

12C2

a(κ − 1)2 +1

2κ2 − 1

12Ca(3 − 4κ + κ2) + κs

We can solve {det(A) = 0, f(κ1) = 0, f(κ2) = 0, |κ1| < 1, |κ2| < 1, κ1 6= κ2} to get

s, κ1, κ2 for different values of Ca. There may exist more than one eigenvalue s and we

compute the largest real part of all the eigenvalues. By using the software Mathematica

we get the result as in Figure 3.2.

−1.5−1.4−1.3−1.2−1.1

−1.0 −0.9−0.8−0.7−0.6−0.5−0.4−0.3−0.2−0.10.0 0.1

−1.6

Third order scheme and the SILW procedure with kd=1

max

{Re(

\tild

e{s}

)}

Ca0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Figure 3.2: GKS analysis for the third order scheme and the SILW procedure with kd = 1

Case 2. If κ1 = κ2 = κ, φj is in the form (3.14). In this case, the numerical boundary

conditions are (3.16) and the linear system Aσ = 0 for {σ1, σ2} has the coefficient matrix

A =

a11 a12

a21 a22

where

a11 = −2

3+

5

12C2

a(κ − 1)2 + 2κ − 1

2κ2 +

5

12Ca(κ

2 − 4κ + 3) + s

a12 = −κ(κ − 2) +5

6Caκ(κ − 2) +

5

6C2

aκ(κ − 1)

a21 = −2

3− 1

12C2

a(κ − 1)2 +1

2κ2 − 1

12Ca(3 − 4κ + κ2) + κs

a22 = −1

6Caκ(κ − 2) − 1

6C2

aκ(κ − 1) + κ(κ + s)

As before, the relevant pair is {s = 0.890809, κ = 0.322185}. In order to obtain non-

trivial eigensolutions, det(A) should be zero. This leads to Ca = −2.31192− 1.49087ı or

Ca = −2.31192+1.49087ı. Since Ca is a real number in [0, 1), there has no eigensolution.

15

From Figure 3.2, we can find that for small values of Ca (which correspond to the

situation that the boundary does not coincide with but is very close to the first grid

point, a typical case in “cut cells”), there exist non-trivial solutions of Re(s) > 0. The

third order scheme and the SILW procedure with kd = 1 is thus not stable for all values

of Ca.

If we perform the same analysis as before with kd = 2, we can find that there exists

no nontrivial eigensolution for all values of Ca.

When we use the SILW procedure to obtain values of ghost points, we always expect

to find (kd)min, the minimum number of derivatives needed to be obtained through the

ILW procedure, which can ensure stability for any position Ca. For the third order

semi-discrete scheme, (kd)min = 2.

Secondly, stability analysis is performed on the left-quarter plane problem

ut + c ux = 0, x ∈ (−∞, b], t ≥ 0, c > 0

u(x, 0) = u0(x), x ∈ (−∞, b](3.17)

For such outflow boundary, values of the ghost points are obtained by classical extrapo-

lation, and they have no relationship with Cb. Similar GKS analysis as performed before

for the inflow case can be applied here. The characteristic equation is still (3.11). The

eigensolution here is {φj(s)}N−∞, hence the condition (3.7) is replaced by

limj→−∞

φj(s) = 0

Now we should focus on the roots of (3.11) which satisfy |κ| > 1. As before, there is

one root which satisfies this property, so we can get φj = σκj. If the values of the ghost

points are obtained by the third order extrapolation

uN+1 = uN−2 − 3uN−1 + 3uN

the numerical boundary condition at j = N is obtained

sφN = −(1

2φN−2 − 2φN−1 +

3

2φN)

16

By φj = σκj , we can see that the numerical boundary condition is

σ(1

2− 2κ + (s +

3

2)κ2) = 0

In order to get nontrivial solution, that is, σ 6= 0, we need that

1

2− 2κ + (s +

3

2)κ2 = 0

f(κ) = 0

Solving this system, we can get {s = 0, κ = 1}. From the previous analysis, we

already know that κ = 1 will transform to |κ| < 1 after perturbation, hence it is not an

eigensolution. So, problem (3.17) with the third order scheme is stable.

In fact, it is proved in [4] that a stable finite difference scheme with outflow extrap-

olation is stable for a linear hyperbolic initial value problem.

3.1.2 Eigenvalue spectrum visualization

Unlike the GKS analysis, which considers the boundary conditions at each end sepa-

rately, the method of eigenvalue spectrum visualization [19] considers stability with the

two boundaries together. Again, in the case of stability analysis, we set g(t) = 0 without

loss of generality.

The semi-discrete schemes can be expressed as a linear system of equations in a

matrix-vector form as

d~U

dt= − c

∆xQ~U (3.18)

where ~U = (u0, u1, · · · , un)T and Q is the coefficient matrix of the spatial discretiza-

tion. This system contains the chosen inner scheme as well as two numerical boundary

conditions.

Let u(x, t) = estv(x), (3.18) changes to

s~U = −Q~U (3.19)

A nontrivial solution ~U satisfying (3.19) is an eigenvector of the matrix −Q and s is

the corresponding eigenvalue. The problem reduces to finding whether there exists any

17

eigenvalue of −Q with Re(s) > 0. As pointed out in [19], we only need to focus on the

eigenvalues which keep O(1) distance from the imaginary axis when the grid number N

increases. Just like in the GKS analysis, there may be more than one such eigenvalues of

the matrix −Q, and we choose the largest real part of all the candidate eigenvalues. We

perform this analysis with the third order scheme and the SILW procedure with kd = 1

as an example, using the Matlab. The result is in Figure 3.3.

−1.6−1.5−1.4−1.3−1.2−1.1

−1.0 −0.9−0.8−0.7−0.6−0.5−0.4−0.3−0.2−0.1

0.00.1


max

{Re(

\tild

e{s}

)} Ca0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Figure 3.3: Eigenspectrum analysis: third order scheme and the SILW procedure withkd = 1

Comparing Figure 3.2 and Figure 3.3, we find that they are almost the same. This

gives us confidence that the eigenspectrum analysis can produce reliable conclusions

about stability, even though it is less mathematically rigorous.

3.2 Fully-discrete schemes

In reality, we always use fully-discrete schemes to solve the problems. This includes

time discretization of the semi-discrete scheme. In this paper, the third order TVD

Runge-Kutta time discretization is used as an example. Firstly, we recall the stability

domain of such time discretization method. Let us consider the following general system

du

dt= F (t, u)

To derive the stability domain, we use F (t, u) = su. This relationship substituted in

18

the third order time discretization leads to the time discrete equation

un+1 = (1 + µ +µ2

2+

µ3

6)un

where un = u(x, tn), µ = s∆t, and ∆t is the time step. Assuming a solution is of the

form un = znu0, here z is a complex number, the stability domain of the method is

|z(µ)| ≤ 1, z(µ) = 1 + µ +µ2

2+

µ3

6(3.20)

Recall that, in the semi-discrete case, an eigensolution is in the form uj(t) = estφj =

esc t

∆x φj with Re(s) ≥ 0. In the fully-discrete scheme, an eigensolution is in the form

un+1j = z(µ)un

j with µ = s c∆t∆x

and |z(µ)| > 1. Here s is an eigenvalue of the semi-discrete

scheme. In both semi-discrete and fully discrete cases, the scheme is unstable if such

candidate eigensolution exists. Take

λcfl =c∆t

∆x

where λcfl > 0. From now on, we would like to verify stability with (λcfl)max, which is

the maximum value of λcfl to ensure stability for the corresponding Cauchy problem. In

other words, we would not want the boundary condition to reduce the CFL number for

stability.

In the periodic case, solutions can be assumed to be uj(t) = 1√2π

u(ω, t)eıωx. In this

circumstance, the third order scheme can be transformed to:

du(ω, t)

dt=

c

∆xu(ω, t)(−1

6e−2ıωh + e−ıωh − 1

2− 1

3eıωh)

=c

∆x((−1

6cos(2ωh) +

2

3cos(ωh) − 1

2) + (

1

6sin(2ωh) − 4

3sin(ωh))ı)u(ω, t)

(3.21)

Compared with F (t, u) = su, we have

s =c

∆x((−1

6cos(2ωh) +

2

3cos(ωh) − 1

2) + (

1

6sin(2ωh) − 4

3sin(ωh))ı)

and we get

µ =c∆t

∆x((−1

6cos(2ωh) +

2

3cos(ωh) − 1

2) + (

1

6sin(2ωh) − 4

3sin(ωh))ı)

= λcfl((−1

6cos(2ωh) +

2

3cos(ωh) − 1

2) + (

1

6sin(2ωh) − 4

3sin(ωh))ı)

(3.22)

19

In order to get stability, µ in (3.22) should satisfy (3.20). By solving the inequality

one can get a range of λcfl. The maximum value of this range is recorded as (λcfl)max.

Because of the algebraic complexity, it is usually difficult to obtain analytically the value

(λcfl)max. Instead, a procedure in Matlab can be used to get (λcfl)max numerically. The

values of (λcfl)max of the different upwind-biased schemes considered in this paper are

listed in Table 3.1.

Table 3.1: (λcfl)max of different schemes

Scheme (λcfl)max

Third order scheme 1.62

Fifth order scheme 1.43

Seventh order scheme 1.24

Ninth order scheme 1.12

Eleventh order scheme 1.04

Thirteenth order scheme 0.99

3.2.1 GKS analysis

For GKS analysis, s is the eigenvalue obtained in the semi-discrete case and

µ = s∆t = (λcfl)maxs

There may exist more than one eigenvalues s, the maximum value of |z| is shown in

Figure 3.4 by the software Mathematica.

20

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

max

{abs

(z)}


Ca0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Figure 3.4: GKS analysis: third order scheme and the SILW procedure with kd = 1

3.2.2 Eigenvalue spectrum visualization

This method is based on the matrix formulation (3.18). After discretizing in time

with the third order Runge-Kutta method, we can get

~Un+1 = (I + (−c∆t

∆xQ) +

1

2(−c∆t

∆xQ)2 +

1

6(−c∆t

∆xQ)3)~Un (3.23)

where I is the identity matrix.

Making use of the analysis of the fully-discrete scheme before, we assume a solution

of the form ~Un = zn ~U0. Substituting this solution into (3.23), we can get

z~Un = G~Un

G = I − c∆t

∆xQ +

1

2(c∆t

∆xQ)2 − 1

6(c∆t

∆xQ)3

where z can be recognized as an eigenvalue with ~Un as the associated eigenvector of

the matrix G. If such eigenvalue with |z| > 1 with an associated non-trivial eigenvector

exists, the scheme is unstable. That is, we need all the eigenvalues of G to lie inside the

unit circle. i.e. |z| ≤ 1, to ensure stability of the fully-discrete approximation.

Result of this analysis for the third order scheme and the SILW procedure with kd = 1

is shown in Figure 3.5.

21

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

max

{abs

(z)}


Ca0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Figure 3.5: Eigenvalue spectrum analysis: Third order scheme and the SILW procedurewith kd = 1.

One can see that Figure 3.4 and Figure 3.5 are almost the same, indicating that both

methods of analysis produce consistent results. The scheme is not stable when Ca is

small.

If we use procedures in Section 3.2.1 and Section 3.2.2 to the third order scheme

and the SILW procedure with kd = 2, there exists no eigensolution, indicating that the

scheme is stable for all Ca.

Next, in Table 3.2, we give the results of the stability analysis, giving (kd)min required

for the SILW inflow boundary treatment for different schemes to remain stable, under

the same (λcfl)max as shown in Table 3.1 for pure initial value problems. Again, the third

order TVD Runge-Kutta method is used for the time discretization.

For the remaining schemes in Section 2.1, results as in Figure 3.5 are shown in Figures

3.6, 3.7 and 3.8. These figures indicate the value ranges of Ca for which the scheme is

unstable with kd just below the stability thresholds listed in Table 3.2.

22

Table 3.2: (kd)min to ensure stability for schemes of different orders

Scheme (kd)min

Third order scheme 2

Fifth order scheme 3

Seventh order scheme 4

Ninth order scheme 6

Eleventh order scheme 8

Thirteenth order scheme 10

0 0.20.40.60.81

1.21.41.61.82

2.22.42.62.83

3.23.43.6

max

{abs

(z)}

Fifth order scheme and the SILW procedure with kd=2

Ca0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

max

{abs

(z)}

Seventh order scheme and the SILW procedure with kd=3

Ca0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Figure 3.6: Results of the fifth order scheme and the SILW procedure with kd = 2 (left)and the seventh order scheme and the SILW procedure with kd = 3 (right).

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

max

{abs

(z)}

Ninth order scheme and the SILW procedure with kd=5

Ca0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

x

max

{abs

(z)}

Eleventh order scheme and the SILW procedure with kd=7

Ca0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Figure 3.7: Results of the ninth order scheme and the SILW procedure with kd = 5 (left)and the eleventh order scheme and the SILW procedure with kd = 7 (right).

23

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

max

{abs

(z)}

Thirteenth order scheme and the SILW procedure with kd=9

Ca0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Figure 3.8: Results of the thirteenth order scheme and the SILW procedure with kd = 9.

4 Numerical examples

In this section we provide numerical examples to demonstrate the stability results

predicted by the analysis in Table 3.2.

4.1 The linear advection equation

The first example is an one-dimensional linear advection equation

ut + ux = 0, x ∈ [−1, 1], t ≥ 0

u(−1, t) = g(t) = 0.25 + 0.5 sin(πt), t ≥ 0

u(x, 0) = 0.25 + 0.5 sin(πx), x ∈ [−1, 1]

(4.24)

The exact solution is

u(x, t) = 0.25 + 0.5 sin(π(x − t))

In order to verify stability, we choose

∆t = (λcfl)max∆x (4.25)

• Third order scheme.

We first consider the third order scheme and the SILW procedure with kd = 1. We

choose Ca = 0.001, i.e. the physical inflow boundary is very close to the first grid point,

a typical situation of “cut cells”. Figure 4.9 (left) shows that the solution has strong

spurious oscillations with very large magnitudes after a short computational time. This

24

clearly demonstrate that the method is unstable, which is consistent with the analysis.

When we take kd = 2, as shown in Figure 4.9 (right), the solution remains stable and

accurate after a very long time simulation, clearly demonstrating the stability of the

scheme predicted by theory. A grid refinement study (not shown here to save space)

verifies the designed third order accuracy.

X-1 -0.5 0 0.5 1

-8E+18

-6E+18

-4E+18

-2E+18

0

2E+18

4E+18Third order schemeExact solution

X-1 -0.5 0 0.5 1

0

0.5

1

Third order schemeExact Solution

Figure 4.9: Numerical results obtained with the third order scheme. Ca = 0.001, Cb = 0.7with N = 40 grid points. Left: kd = 1, t = 10; Right: kd = 2, t = 10000.

• Fifth order scheme.

Next, we consider the fifth order scheme and the SILW procedure with kd = 2.

Figure 4.10 (left) shows that the solution has strong spurious oscillations with very large

magnitudes after a short computational time. This clearly demonstrate that the method

is unstable, which is consistent with the analysis. When we take kd = 3, as shown

in Figure 4.10 (right), the solution remains stable and accurate after a very long time

simulation, clearly demonstrating the stability of the scheme predicted by theory. Again,

the result of a grid refinement study to verify accuracy is not shown to save space.

25

X-1 -0.5 0 0.5 1

-6E+25

-4E+25

-2E+25

0

2E+25

4E+25

Fifth order schemeExact solution

x-1 -0.5 0 0.5 1

0

0.5

1

Fifth order schemeExact Solution

Figure 4.10: Numerical results obtained with the fifth order scheme. Ca = 0.75, Cb = 0.7with N = 40 grid points. Left: kd = 2, t = 4; Right: kd = 3, t = 10000.

• Seventh order scheme.

We repeat our numerical experiment with the seventh order scheme. With kd = 3,

Figure 4.11 (left) clearly shows instability. When we increase the terms using the ILW

procedure to kd = 4, the scheme becomes stable as shown in Figure 4.11 (right), which

is consistent with the analysis.

X-1 -0.5 0 0.5 1

-4E+08

-2E+08

0

2E+08

Seventh order schemeExact solution

X-1 -0.5 0 0.5 1

0

0.5

1

Seventh order schemeExact solution

Figure 4.11: Numerical results obtained with the seventh order scheme. Ca = 0.001,Cb = 0.7 with N = 40 grid points. Left: kd = 3, t = 10; Right: kd = 4, t = 10000.

• Ninth order scheme.

26

The simulation for the ninth order scheme again verifies our analysis. Figure 4.12

(left) shows instability with kd = 5, and Figure 4.12 (right) shows stability with kd = 6.

X-1 -0.5 0 0.5 1

-2E+15

0

2E+15

4E+15Ninth order schemeExact solution

X-1 -0.5 0 0.5 1

0

0.5

1

Ninth order schemeExact solution

Figure 4.12: Numerical results obtained with the ninth order scheme. Ca = 0.25, Cb =0.7 with N = 40 grid points. Left: kd = 5, t = 20; Right: kd = 6, t = 10000.

• Eleventh order scheme.

The simulation is repeated for the eleventh order scheme with kd = 7 (Figure 4.13

left) showing instability and with kd = 8 (Figure 4.13 right) showing stability.

X-1 -0.5 0 0.5 1

-2E+09

-1E+09

0

1E+09

2E+09

3E+09

4E+09Eleventh order schemeExact solution

X-1 -0.5 0 0.5 1

0

0.5

1

Eleventh order schemeExact solution

Figure 4.13: Numerical results obtained with the eleventh order scheme. Ca = 0.17,Cb = 0.7 with N = 40 grid points. Left: kd = 7, t = 20; Right: kd = 8, t = 10000.

• Thirteenth order scheme

27

Finally, the simulation is performed for the thirteenth order scheme with kd = 9

(Figure 4.14 left) showing instability and with kd = 10 (Figure 4.14 right) showing

stability.

X-1 -0.5 0 0.5 1

-400000

-200000

0

200000

Thirteenth order schemeExact solution

X-1 -0.5 0 0.5 1

0

0.5

1

Thirteenth order schemeExact solution

Figure 4.14: Numerical results obtained with the thirteenth order scheme. Ca = 0.001,Cb = 0.7 with N = 40 grid points. Left: kd = 9, t = 40; Right: kd = 10, t = 10000.

4.2 Burgers equation

In this subsection we pay our attention to the nonlinear scalar Burgers equation

ut + (u2

2)x = 0, x ∈ [0, 2π], t ≥ 0

u(0, t) = g(t), t ≥ 0

u(x, 0) = u0(x), x ∈ [0, 2π]

(4.26)

where u0(x) is the initial condition. We assume u(0, t) > 0, hence x = 0 is an inflow

boundary and g(t) is the prescribed boundary condition. We take g(t) = u(0, t), where

u(x, t) is the exact solution of the initial value problem on (0, 2π) with periodic boundary

condition for all t.

Two examples are given in this section. For simplicity, we will only test the third

order scheme and the fifth order scheme as the inner schemes.

28

4.2.1 Example 1

We take the initial condition as

u0(x) = 1 + 0.5 sin(x) (4.27)

At t = 1.0, we have a smooth solution. Values of the ghost points near the inflow

boundary are obtained by the SILW method. When considering ghost points near the

outflow boundary, we use extrapolation in the appropriate order. From the previous

analysis, if we use the fifth order scheme to approximate the spatial derivative, we need

kd = 3 in the SILW procedure to ensure stability for all values of Ca. In this example,

it appears that kd = 2 is enough to ensure stability for all values of Ca. The numerical

results are summarized in Tables 4.3 and 4.4.

Firstly, we use

∆t =λcfl ∆x

α(4.28)

to verify stability.

Secondly, we use

∆t =λcfl (∆x)

53

α(4.29)

to verify the designed order of accuracy.

In numerical examples, we always set λcfl = (λcfl)max, and α = max0≤i≤N

{|uni |} where

uni is the numerical solution at time level n and grid xi.

Table 4.3 shows that the fifth order scheme using the SILW procedure with kd = 2 is

stable for this example, under the maximum CFL number for the inner scheme, for Ca =

0.75 and Ca = 0.40. Of course, only third order accuracy can be obtained asymptotically,

restricted by the time discretization accuracy. Even though we do not list the results,

stability has also been observed with kd = 2 for other values of Ca. This appears to

be better than what our analysis indicates before, which would predict that kd = 3 is

needed for stability over the whole range of Ca. Besides nonlinearity, the main reason

might be that the solution varies quite a lot over the computational domain, and the

29

Table 4.3: Fifth order scheme with kd = 2, Cb = 0.7 at t = 1.0 for (4.27) with (4.28)

N Ca = 0.75 Ca = 0.40

L2 error order L∞ error order L2 error order L∞ error order

40 1.481E-03 – 1.892E-03 – 1.461E-03 – 1.871E-03 –

80 2.069E-04 2.839 3.054E-04 2.631 2.100E-04 2.799 3.134E-04 2.578

160 2.892E-05 2.839 4.598E-05 2.732 2.878E-05 2.867 4.577E-05 2.776

320 3.632E-06 2.993 5.841E-06 2.977 3.643E-06 2.982 5.865E-06 2.964

640 4.656E-07 2.964 7.569E-07 2.948 4.669E-07 2.964 7.596E-07 2.949

1280 5.837E-08 2.996 9.507E-08 2.993 5.834E-08 3.001 9.500E-08 2.999

solution near the inflow boundary x = 0, which is of the size max0≤t≤1 |g(t)|, is much less

than α = max |un|. Hence the effective CFL limit near the boundary x = 0 is less than

(λcfl)max. Even though we have shown before that, under the maximum CFL condition

for the inner scheme λcfl = 1.43, we would need kd = 3 to ensure stability for the whole

range of Ca, we can also show that, if we allow a smaller CFL number λcfl ≤ 1.02, then

kd = 2 is enough to ensure stability for the whole range of Ca. We can verify that, under

the time step restriction (4.28), max0≤t≤1 |g(t)|∆t∆x

is indeed less than 1.02. Hence the

numerical stability in this case is not surprising. This example shows that sometimes

stability is better for nonlinear problems than for linear problems.

Clearly, if we use the time step (4.29), we can observe the designed fifth order of

accuracy in Table 4.4.

4.2.2 Example 2

We now consider a different initial condition

u0(x) = 1 + 0.2 sin(x) (4.30)

The difference from the previous initial condition is that |un| does not vary as much when

comparing its value near the inflow boundary x = 0 and over the whole computational

30


N Ca = 0.75 Ca = 0.40


40 2.430E-04 – 3.925E-04 – 2.523E-04 – 3.862E-04 –

80 1.096E-05 4.470 2.313E-05 4.085 1.118E-05 4.495 2.318E-05 4.058

160 3.866E-07 4.826 8.304E-07 4.800 3.908E-07 4.839 8.457E-07 4.777

320 1.262E-08 4.937 2.752E-08 4.915 1.269E-08 4.944 2.773E-08 4.931

640 4.016E-10 4.974 8.787E-10 4.969 4.027E-10 4.978 8.816E-10 4.975

1280 1.265E-11 4.989 2.770E-11 4.987 1.267E-11 4.991 2.774E-11 4.990

domain.


Firstly, we use the SILW procedure with kd = 1. We set the final time t = 1

and Ca = 0.001, and observe that errors in this case become lager as ∆x is decreased,

implying numerical instability. Next we take kd = 2 and show the results in Table 4.5.

We can clearly see that the scheme is stable in this case and the designed third order

accuracy is obtained.

Table 4.5: Third order scheme with kd = 2, Cb = 0.7 at t = 1.0 for (4.30) with (4.28)

N Ca = 0.001 Ca = 0.35


40 4.616E-04 – 4.084E-04 – 4.733E-04 – 4.042E-04 –

80 6.423E-05 2.845 5.805E-05 2.815 6.573E-05 2.848 5.721E-05 2.821

160 8.589E-06 2.903 7.812E-06 2.893 8.884E-06 2.887 7.801E-06 2.874

320 1.087E-06 2.982 9.901E-07 2.980 1.124E-06 2.982 9.883E-07 2.981

640 1.373E-07 2.985 1.252E-07 2.983 1.421E-07 2.984 1.250E-07 2.983

1280 1.730E-08 2.988 1.579E-08 2.987 1.792E-08 2.987 1.578E-08 2.985

31


Firstly, we use the SILW procedure with kd = 2. We choose t = 1, and Ca = 0.75,

and observe that the errors become lager as ∆x is decreased when the time step is taken

according to (4.28), indicating instability. Next, we increase kd to kd = 3. Stable and

accurate results are then obtained, summarized in Tables 4.6 and 4.7 for the time steps

taken according to (4.28) and (4.29) respectively.


N Ca = 0.75 Ca = 0.40


40 2.212E-04 – 2.004E-04 – 2.283E-04 – 2.047E-04 –

80 3.156E-05 2.809 2.905E-05 2.786 3.173E-05 2.847 2.918E-05 2.810

160 4.084E-06 2.950 3.778E-06 2.943 4.117E-06 2.946 3.806E-06 2.939

320 5.293E-07 2.948 4.903E-07 2.946 5.296E-07 2.958 4.906E-07 2.955

640 6.653E-08 2.992 6.169E-08 2.990 6.664E-08 2.991 6.179E-08 2.989

1280 8.394E-09 2.987 7.787E-09 2.986 8.404E-09 2.987 7.796E-09 2.987


N Ca = 0.75 Ca = 0.40


40 8.062E-06 – 7.695E-06 – 8.466E-06 – 8.151E-06 –

80 2.921E-07 4.786 2.887E-07 4.736 2.956E-07 4.840 2.936E-07 4.795

160 9.674E-09 4.916 9.634E-09 4.905 9.760E-09 4.921 9.749E-09 4.913

320 3.114E-10 4.957 3.106E-10 4.955 3.119E-10 4.968 3.123E-10 4.964

640 9.869E-12 4.980 9.852E-12 4.978 9.858E-12 4.984 9.879E-12 4.982

1280 3.095E-13 4.995 3.119E-13 4.981 3.081E-13 5.000 3.129E-13 4.981

32

4.3 Euler equations

The analysis performed in this paper can be easily generalized to linear hyperbolic

systems. The numerical procedure works well also for nonlinear hyperbolic conservation

laws, such as Euler equations of compressible gas dynamics

~Ut + ~F (~U)x = 0, x ∈ [a, b], t > 0 (4.31)

where the conservative variables are given by

~U =

u1

u2

u3

=

ρ

ρu

E

and the flux is given by

~F (~U) =

u2

(γ − 1)u3 + 3−γ

2

u22

u1

(γu3 − γ−12

u22

u1)u2

u1

where ρ is density, u is velocity, E = 12ρu2 + ρe represents the total energy and p is the

pressure. The equation of state is e = p

(γ−1)ρ. Here we choose γ = 1.4.

We take a = −π and b = π in (4.31) and the initial condition

ρ0(x) = 1 + 0.2 sin(x), x ∈ [−π, π], t ≥ 0

u0(x) = 2, x ∈ [−π, π]

p0(x) = 2, x ∈ [−π, π]

The exact solution with periodic boundary condition is

ρ(x, t) = 1 + 0.2 sin(x − 2t), x ∈ [−π, π], t ≥ 0

u(x, t) = 2, x ∈ [−π, π], t ≥ 0

p(x, t) = 2, x ∈ [−π, π], t ≥ 0

33

The three eigenvalues of the Jacobian matrix ~F ′(~U) are u−c, u, u+c where c =√

γp

ρ

is the sound speed. Since c < 2 = u, the three eigenvalues are all greater than 0 in the

whole domain (x, t) ∈ [−π, π] × [0, +∞). x = −π is an inflow boundary and x = π

is an outflow boundary for the three variables. The boundary condition for the inflow

boundary in the conservative variables is

u1(−π, t) =g1(t) = 1 + 0.2 sin(2t), t ≥ 0

u2(−π, t) =g2(t) = 2 + 0.4 sin(2t), t ≥ 0

u3(−π, t) =g3(t) = 2 + 0.4 sin(2t) +2

γ − 1, t ≥ 0

The details of the ILW and SILW procedures for hyperbolic systems are obtained

along the same lines as those for the scalar equations, but are algebraically more com-

plicated. We omit the derivation details here to save space and refer to [16, 18]. We use

the third order scheme and the fifth order scheme to solve (4.31).


Firstly, we use SILW procedure with kd = 1. We choose t = 1 and Ca = 0.001, and

observe that the errors become lager as ∆x decreases, indicating instability. Results of

the third order scheme and the SILW procedure with kd = 2 are stable and accurate,

and are given in Table 4.8.

Table 4.8: Third order scheme with kd = 2, Cb = 0.7 at t = 1.0 for (4.31) with (4.28).

N Ca = 0.001 Ca = 0.35


40 2.223E-04 – 1.919E-04 – 2.763E-04 – 1.869E-04 –

80 2.860E-05 2.958 2.447E-05 2.971 3.616E-05 2.934 2.417E-05 2.952

160 3.646E-06 2.972 3.085E-06 2.988 4.622E-06 2.968 3.064E-06 2.980

320 4.601E-07 2.986 3.866E-07 2.996 5.841E-07 2.984 3.854E-07 2.991

640 5.778E-08 2.993 4.837E-08 2.999 7.339E-08 2.993 4.830E-08 2.996

1280 7.245E-09 2.996 6.052E-09 2.999 9.200E-09 2.996 6.046E-09 2.998

34


Firstly, we use the SILW procedure with kd = 2. We choose t = 1 and Ca = 0.75,

and observe that the errors become lager as ∆x decreases, implying instability.

Results of the fifth order scheme and the SILW procedure with kd = 3, under two

different choices of time steps, are given in Tables 4.9 and 4.10 respectively. Table 4.9

shows that the fifth order scheme and the SILW procedure with kd = 3 is stable under the

maximum allowable CFL number for the inner scheme. Table 4.10 shows the designed

fifth order of accuracy when smaller time steps are taken.

Table 4.9: Fifth order scheme with kd = 3, Cb = 0.7 at t = 1.0 for (4.31) with (4.28).

N Ca = 0.75 Ca = 0.40


40 3.454E-05 – 2.351E-05 – 3.497E-05 – 2.390E-05 –

80 4.544E-06 2.926 3.096E-06 2.924 4.593E-06 2.928 3.130E-06 2.933

160 5.772E-07 2.977 3.942E-07 2.974 5.821E-07 2.980 3.972E-07 2.978

320 7.325E-08 2.978 5.007E-08 2.977 7.359E-08 2.984 5.027E-08 2.982

640 9.230E-09 2.988 6.311E-09 2.988 9.253E-09 2.992 6.325E-09 2.991

1280 1.159E-09 2.994 7.923E-10 2.994 1.159E-09 2.997 7.928E-10 2.996

Table 4.10: Fifth order scheme with kd = 3, Cb = 0.7 at t = 1.0 for (4.31) with (4.29).

N Ca = 0.75 Ca = 0.40


40 1.564E-06 – 1.136E-06 – 1.811E-06 – 1.186E-06 –

80 5.397E-08 4.857 3.981E-08 4.835 5.904E-08 4.939 4.069E-08 4.865

160 1.777E-09 4.924 1.309E-09 4.927 1.898E-09 4.959 1.359E-09 4.904

320 5.701E-11 4.962 4.199E-11 4.962 6.023E-11 4.978 4.466E-11 4.928

640 1.811E-12 4.976 1.711E-12 4.617 1.665E-12 5.177 1.710E-12 4.707

35

5 Concluding remarks

In this paper, we have discussed stability of high order upwind-biased finite differ-

ence schemes for solving hyperbolic conservation laws on a finite domain. Values of ghost

points near the inflow boundary are obtained by an inverse Lax-Wendroff (ILW) or a

simplified ILW (SILW) procedure and values of ghost points near the outflow boundary

are obtained by a classical Lagrangian extrapolation of appropriate order. The outflow

extrapolation and inflow ILW boundary treatments have both been proved to maintain

stability and the order of accuracy, with the same CFL number (λcfl)max as the peri-

odic case, for any of the high order schemes presented and for any Ca, which measures

the distance between the physical boundary and the closet grid point. For the SILW

boundary treatment, stability depends on kd, which is the number of terms obtained

by the ILW procedure. SILW maintains stability with (λcfl)max as the periodic case for

any Ca when kd is at least taken as (kd)min, whose values for various orders of schemes

are summarized in Table 3.2 by our stability analysis. Numerical examples are provided

to demonstrate stability or instability predicted by the analysis in Table 3.2. Stability

analysis in this paper is performed based on both the GKS analysis and eigenvalue spec-

trum visualization. The GKS analysis breaks the problem into three simpler problems,

the inner problem and the two quarter-plane problems corresponding to the two types

of boundaries. It is mathematically rigorous but could be algebraically highly compli-

cated, especially for high order schemes. The eigenvalue spectrum visualization method

transforms the scheme into a matrix form, containing both boundary conditions. It

can be relatively easily performed for high order methods. It appears that, when both

methods are applied, they produce consistent prediction for stability. The results of this

paper are expected to provide valuable guidance to users of high order upwind-biased

finite difference schemes, such as WENO schemes, for solving hyperbolic problems with

the SILW boundary treatment procedure. Even though the analysis is performed in one-

dimension, the stability prediction is expected to be valid for multi-dimensional problems.

36

The SILW boundary treatment is particularly attractive for multi-dimensional problems

with complex geometry of the computational domain, including moving geometry, when

simulated by a finite difference method on a fixed Cartesian mesh, see [16, 17, 18] for

numerical experiment examples.

References

[1] D.S. Balsara and C.-W. Shu, Monotonicity preserving weighted essentially non-

oscillatory schemes with increasingly high order of accuracy, Journal of Computa-

tional Physics, 160:405–452, 2000.

[2] M.J. Berger, C. Helzel and R.J. Leveque, h-box methods for the approximation of hy-

perbolic conservation laws on irregular grids, SIAM Journal on Numerical Analysis,

41:893–918, 2003.

[3] M.H. Carpenter, D. Gottlieb, S. Abarbanel and W.-S. Don, The theoretical accuracy

of Runge-Kutta time discretizations for the initial boundary value problem: a study

of the boundary error. SIAM Journal on Scientific Computing, 16:1241-1252, 1995.

[4] M. Goldberg, On a boundary extrapolation theorem by Kreiss. Mathematics of Com-

putation, 31:469-477, 1977.

[5] M. Goldberg and E. Tadmor, Scheme-independent stability criteria for difference ap-

proximations of hyperbolic initial-boundary value problems. I, Mathematics of Com-

putation, 32:1097–1107, 1978.

[6] M. Goldberg and E. Tadmor, Scheme-independent stability criteria for difference

approximations of hyperbolic initial-boundary value problems. II. Mathematics of

Computation, 36:603–626, 1981.

37

[7] B. Gustafsson, H.-O. Kreiss and A. Sundstrom, Stability theory of difference approx-

imations for mixed initial boundary value problem. II, Mathematics of Computation,

26:649–686, 1972.

[8] W.D. Henshaw, A high-order accurate parallel solver for Maxwell’s equations on

overlapping grids, SIAM Journal on Scientific Computing, 28:1730–1765, 2006.

[9] W.D. Henshaw, H.-O. Kreiss and L.G.M. Reyna, A fourth-order accurate difference

approximation for the incompressible Navier-Stokes equations, Computers & Fluids,

23:575–593, 1994.

[10] L. Huang, C.-W. Shu and M. Zhang, Numerical boundary conditions for the fast

sweeping high order WENO methods for solving the Eikonal equation, Journal of

Computational Mathematics, 26:336–346, 2008.

[11] G. Jiang and C.-W. Shu, Efficient implementation of weighted ENO schemes. Jour-

nal of Computational Physics, 126:202-228, 1996.

[12] X.-D. Liu, S. Osher and T. Chan, Weighted essentially nonoscillatory schemes.

Journal of Computational Physics, 115:200-212, 1994.

[13] C.-W. Shu, High order weighted essentially nonoscillatory schemes for convection

dominated problems, SIAM Review, 51:82–126, 2009.

[14] C.-W. Shu and S. Osher, Efficient implementation of essentially non-oscillatory

shock-capturing schemes. Journal of Computational Physics, 77:439-471, 1988.

[15] J.C. Strikwerda, Initial boundary value problems for the method of lines. Journal of

Computational Physics, 34:94–107, 1980.

[16] S. Tan and C.-W. Shu, Inverse Lax-Wendroff procedure for numerical boundary

conditions of conservation laws. Journal of Computational Physics, 229:8144–8166,

2010.

38

[17] S. Tan and C.-W. Shu, A high order moving boundary treatment for compressible

inviscid flows, Journal of Computational Physics, 230:6023–6036, 2011.

[18] S. Tan, C. Wang, C.-W. Shu and J. Ning, Efficient implementation of high order

inverse Lax-Wendroff boundary treatment for conservation laws. Journal of Compu-

tational Physics, 231:2510–2527, 2012.

[19] F. Vilar and C.-W. Shu, Development and stability analysis of the inverse Lax-

Wendroff boundary treatment for central compact schemes. Mathematical Modelling

and Numerical Analysis, 49:39–67, 2015.

39

Abstract - Brown University · This is why the schemes are called “upwind-biased”. Also notice...

Documents

Transcript of Abstract - Brown University · This is why the schemes are called “upwind-biased”. Also notice...